on the probability distribution of stock returns in the mike-farmer model

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1、OntheprobabilitydistributionofstockreturnsintheMike-FarmermodelGao-FengGua,b,Wei-XingZhoua,b,c,d,∗aSchoolofBusiness,EastChinaUniversityofScienceandTechnology,Shanghai200237,ChinabSchoolofScience,EastChinaUniversityofScienceandTechnology,Shanghai200237,ChinacResearchCenterforEconophysi

2、cs,EastChinaUniversityofScienceandTechnology,Shanghai200237,ChinadResearchCenterofSystemsEngineering,EastChinaUniversityofScienceandTechnology,Shanghai200237,ChinaAbstractRecently,MikeandFarmerhaveconstructedaverypowerfulandrealisticbehavioralmodeltomimickthedynamicprocessofstockprice

3、formationbasedontheempiricalregularitiesoforderplacementandcancelationinapurelyorder-drivenmarket,whichcansuccess-fullyreproducethewholedistributionofreturns,notonlythewell-knownpower-lawtails,togetherwithseveralotherimportantstylizedfacts.TherearethreekeyingredientsintheMike-Farmer(M

4、F)model:thelongmemoryofordersignscharacterizedbytheHurstindexHs,thedistributionofrelativeorderpricesxinreferencetothesamebestpricedescribedbyaStudentdistribution(orTsallis'q-Gaussian),andthedynamicsofordercancelation.arXiv:0805.3593v1[q-fin.ST]23May2008Theyshowedthatdifferentvaluesoft

5、heHurstindexHsandthefreedomdegreeαxoftheStudentdistributioncanalwaysproducepower-lawtailsinthereturndistributionf(r)withdifferenttailexponentαr.Inthispaper,westudytheoriginofthepower-lawtailsofthereturndistributionf(r)intheMFmodel,basedonextensivesimulationswithdifferentcombinationsof

6、theleftpartfL(x)forx<0andtherightpartfR(x)forx>0off(x).Wefindthatpower-lawtailsappearonlywhenfL(x)hasapower-lawtail,nomatterfR(x)hasapower-lawtailornot.Inaddition,wefindthatthedistributionsofreturnsintheMFmodelatdifferenttimescalescanbewellmodeledbytheStudentdistributions,whosetailexpon

7、entsareclosetothewell-knowncubiclawandincreasewiththetimescale.Keywords:Econophysics;Mike-Farmermodel;Power-lawtail;q-Gaussian;Order-drivenmarketPreprintsubmittedtoPhysicaA2December20081IntroductionManystylizedfactshavebeenunveiledindifferentstockmarkets[1,2,3].Un-derstandingtheunderl

8、yingr